3.2201 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=342 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (b e g-8 c d g+6 c e f)}{e^2 (d+e x)^2 (2 c d-b e)}-\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-8 c d g+6 c e f)}{3 e^2 (2 c d-b e)}-\frac{5 (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-8 c d g+6 c e f)}{8 e}-\frac{5 (2 c d-b e)^2 (b e g-8 c d g+6 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 \sqrt{c} e^2} \]

[Out]

(-5*(6*c*e*f - 8*c*d*g + b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(8*e) - (5*c*(6*c*e*f -
8*c*d*g + b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(2*c*d - b*e)) - (2*(6*c*e*f - 8*c*d*g +
b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(e^2*(2*c*d - b*e)*(d + e*x)^2) - (2*(e*f - d*g)*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d - b*e)*(d + e*x)^4) - (5*(2*c*d - b*e)^2*(6*c*e*f - 8*c*d*g +
b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(16*Sqrt[c]*e^2)

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Rubi [A]  time = 0.597807, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {792, 662, 664, 612, 621, 204} \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4 (2 c d-b e)}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (b e g-8 c d g+6 c e f)}{e^2 (d+e x)^2 (2 c d-b e)}-\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-8 c d g+6 c e f)}{3 e^2 (2 c d-b e)}-\frac{5 (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-8 c d g+6 c e f)}{8 e}-\frac{5 (2 c d-b e)^2 (b e g-8 c d g+6 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 \sqrt{c} e^2} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

(-5*(6*c*e*f - 8*c*d*g + b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(8*e) - (5*c*(6*c*e*f -
8*c*d*g + b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(2*c*d - b*e)) - (2*(6*c*e*f - 8*c*d*g +
b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(e^2*(2*c*d - b*e)*(d + e*x)^2) - (2*(e*f - d*g)*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d - b*e)*(d + e*x)^4) - (5*(2*c*d - b*e)^2*(6*c*e*f - 8*c*d*g +
b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(16*Sqrt[c]*e^2)

Rule 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2
)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rule 664

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[(p*(2*c*d - b*e))/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*
(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a
*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac{(6 c e f-8 c d g+b e g) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx}{e (2 c d-b e)}\\ &=-\frac{2 (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac{(5 c (6 c e f-8 c d g+b e g)) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{d+e x} \, dx}{e (2 c d-b e)}\\ &=-\frac{5 c (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}-\frac{2 (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac{(5 c (6 c e f-8 c d g+b e g)) \int \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{2 e}\\ &=-\frac{5 (6 c e f-8 c d g+b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e}-\frac{5 c (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}-\frac{2 (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac{\left (5 (2 c d-b e)^2 (6 c e f-8 c d g+b e g)\right ) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{16 e}\\ &=-\frac{5 (6 c e f-8 c d g+b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e}-\frac{5 c (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}-\frac{2 (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac{\left (5 (2 c d-b e)^2 (6 c e f-8 c d g+b e g)\right ) \operatorname{Subst}\left (\int \frac{1}{-4 c e^2-x^2} \, dx,x,\frac{-b e^2-2 c e^2 x}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{8 e}\\ &=-\frac{5 (6 c e f-8 c d g+b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e}-\frac{5 c (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}-\frac{2 (6 c e f-8 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^4}-\frac{5 (2 c d-b e)^2 (6 c e f-8 c d g+b e g) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 \sqrt{c} e^2}\\ \end{align*}

Mathematica [A]  time = 2.38693, size = 309, normalized size = 0.9 \[ \frac{2 ((d+e x) (c (d-e x)-b e))^{5/2} \left (e^3 (e f-d g) (b e-c d+c e x)^3-\frac{e^{5/2} \sqrt{d+e x} (b e g-8 c d g+6 c e f) \left (\sqrt{c} \sqrt{e} \sqrt{d+e x} \sqrt{\frac{b e-c d+c e x}{b e-2 c d}} \left (33 b^2 e^2+2 b c e (13 e x-53 d)+4 c^2 \left (22 d^2-9 d e x+2 e^2 x^2\right )\right )+15 \sqrt{e (2 c d-b e)} (b e-2 c d)^2 \sin ^{-1}\left (\frac{\sqrt{c} \sqrt{e} \sqrt{d+e x}}{\sqrt{e (2 c d-b e)}}\right )\right )}{48 \sqrt{c} \sqrt{\frac{b e-c d+c e x}{b e-2 c d}}}\right )}{e^5 (d+e x)^3 (2 c d-b e) (b e-c d+c e x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(e^3*(e*f - d*g)*(-(c*d) + b*e + c*e*x)^3 - (e^(5/2)*(6*c*e*f - 8*
c*d*g + b*e*g)*Sqrt[d + e*x]*(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x]*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]*(33*b^
2*e^2 + 2*b*c*e*(-53*d + 13*e*x) + 4*c^2*(22*d^2 - 9*d*e*x + 2*e^2*x^2)) + 15*Sqrt[e*(2*c*d - b*e)]*(-2*c*d +
b*e)^2*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]]))/(48*Sqrt[c]*Sqrt[(-(c*d) + b*e + c*e*x)
/(-2*c*d + b*e)])))/(e^5*(2*c*d - b*e)*(d + e*x)^3*(-(c*d) + b*e + c*e*x)^2)

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Maple [B]  time = 0.016, size = 4987, normalized size = 14.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x)

[Out]

-15/8*e^7*c/(-b*e^2+2*c*d*e)^3*b^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+
d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d*g-75*e^5*c^3/(-b*e^2+2*c*d*e)^3*b^3/(c*e^2)^(1/2)*arctan((c*e^
2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^3*g+150*e^4*c
^4/(-b*e^2+2*c*d*e)^3*b^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*
e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^4*g+45*e^4*c^3/(-b*e^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e
)*(x+d/e))^(1/2)*x*d^2*g-45*e^5*c^3/(-b*e^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x
*d*f+75*e^6*c^3/(-b*e^2+2*c*d*e)^3*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(
-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^2*f+15/2*e^6*c^2/(-b*e^2+2*c*d*e)^3*b^3*(-(x+d/e)^2*c*e^2+
(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*f+45/2*e^4*c^2/(-b*e^2+2*c*d*e)^3*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+
d/e))^(1/2)*d^2*g-10/3*g*e^2*c^2/(-b*e^2+2*c*d*e)^2*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x-60*e
^3*c^5/(-b*e^2+2*c*d*e)^3*d^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*f+40*e^2*c^4/(-b*e^2+2*c*d*e
)^3*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*g-40*e^3*c^4/(-b*e^2+2*c*d*e)^3*d*(-(x+d/e)^2*c*e^
2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*f-25/2*g*e^4*c^2/(-b*e^2+2*c*d*e)^2*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)
*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^2+15/2*g*e^3*c^2/(-b*
e^2+2*c*d*e)^2*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d+25*g*e^3*c^3/(-b*e^2+2*c*d*e)^2*b^2/(
c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e
))^(1/2))*d^3-15*g*e^2*c^3/(-b*e^2+2*c*d*e)^2*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d^2+25/8*g
*e^5*c/(-b*e^2+2*c*d*e)^2*b^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^
2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d-25*g*e^2*c^4/(-b*e^2+2*c*d*e)^2*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2
)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^4-150*e^5*c^4/(-b*e^
2+2*c*d*e)^3*b^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e
^2+2*c*d*e)*(x+d/e))^(1/2))*d^3*f-90*e^3*c^4/(-b*e^2+2*c*d*e)^3*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^
(1/2)*x*d^3*g+90*e^4*c^4/(-b*e^2+2*c*d*e)^3*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d^2*f+75/4*e
^6*c^2/(-b*e^2+2*c*d*e)^3*b^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^
2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^2*g-12/e^2*c/(-b*e^2+2*c*d*e)^2/(x+d/e)^3*(-(x+d/e)^2*c*e^2+(-b*e^2
+2*c*d*e)*(x+d/e))^(7/2)*f+2/e^5/(-b*e^2+2*c*d*e)/(x+d/e)^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)*
d*g+16/3*g/e^2*c/(-b*e^2+2*c*d*e)^2/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)-5/3*g*e^2*c/(-
b*e^2+2*c*d*e)^2*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)-5/16*g*e^6/(-b*e^2+2*c*d*e)^2*b^5/(c*e^
2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(
1/2))+32*e*c^3/(-b*e^2+2*c*d*e)^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*d*g+10*e^4*c^2/(-b*e^2+2*c
*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*f+15/4*e^6*c/(-b*e^2+2*c*d*e)^3*b^4*(-(x+d/e)^2*
c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*f+16/3*g*c^2/(-b*e^2+2*c*d*e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/
e))^(5/2)+2*g/e^4/(-b*e^2+2*c*d*e)/(x+d/e)^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)-5/8*g*e^4/(-b*e
^2+2*c*d*e)^2*b^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)-2/e^4/(-b*e^2+2*c*d*e)/(x+d/e)^4*(-(x+d/e)
^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)*f-32*c^2/(-b*e^2+2*c*d*e)^3/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d
*e)*(x+d/e))^(7/2)*f-32*e^2*c^3/(-b*e^2+2*c*d*e)^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*f-30*e^3*
c^4/(-b*e^2+2*c*d*e)^3*d^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*f+60*e^2*c^6/(-b*e^2+2*c*d*e)^3
*d^6/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*
(x+d/e))^(1/2))*g-60*e^3*c^6/(-b*e^2+2*c*d*e)^3*d^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*
d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*f+60*e^2*c^5/(-b*e^2+2*c*d*e)^3*d^4*(-(x+d/e)^2
*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*g-15/2*g*e^2*c^2/(-b*e^2+2*c*d*e)^2*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c
*d*e)*(x+d/e))^(1/2)*d^2+32/e*c^2/(-b*e^2+2*c*d*e)^3/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/
2)*d*g-10*e^3*c^2/(-b*e^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*d*g-20*e^3*c^3/(-b*
e^2+2*c*d*e)^3*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*b*f-45/2*e^5*c^2/(-b*e^2+2*c*d*e)^3*b^3*(-(
x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d*f+15/8*e^8*c/(-b*e^2+2*c*d*e)^3*b^5/(c*e^2)^(1/2)*arctan((c*e
^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*f+20*e^4*c^3/(
-b*e^2+2*c*d*e)^3*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*f+12/e^3*c/(-b*e^2+2*c*d*e)^2/(x+d/e)^
3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)*d*g+5*g*e*c^3/(-b*e^2+2*c*d*e)^2*d^3*(-(x+d/e)^2*c*e^2+(-b
*e^2+2*c*d*e)*(x+d/e))^(1/2)*b+10*g*e*c^5/(-b*e^2+2*c*d*e)^2*d^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2
*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))+10*g*e*c^4/(-b*e^2+2*c*d*e)^2*d^3*
(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x+20/3*g*e*c^3/(-b*e^2+2*c*d*e)^2*d*(-(x+d/e)^2*c*e^2+(-b*e^
2+2*c*d*e)*(x+d/e))^(3/2)*x+10/3*g*e*c^2/(-b*e^2+2*c*d*e)^2*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2
)*b-5/4*g*e^4*c/(-b*e^2+2*c*d*e)^2*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x+15/4*g*e^3*c/(-b*e^
2+2*c*d*e)^2*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d+20*e^2*c^3/(-b*e^2+2*c*d*e)^3*d^2*(-(x+d/
e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*b*g-75/4*e^7*c^2/(-b*e^2+2*c*d*e)^3*b^4/(c*e^2)^(1/2)*arctan((c*e^2
)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d*f-15/2*e^5*c^2
/(-b*e^2+2*c*d*e)^3*b^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d*g-150*e^3*c^5/(-b*e^2+2*c*d*e)^3
*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x
+d/e))^(1/2))*d^5*g+150*e^4*c^5/(-b*e^2+2*c*d*e)^3*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c
*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^4*f-20*e^3*c^3/(-b*e^2+2*c*d*e)^3*b*(-(x+d/e
)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*d*g+45*e^4*c^3/(-b*e^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*
c*d*e)*(x+d/e))^(1/2)*d^2*f+30*e^2*c^4/(-b*e^2+2*c*d*e)^3*d^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2
)*b*g-15/4*e^5*c/(-b*e^2+2*c*d*e)^3*b^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d*g-45*e^3*c^3/(-b*e
^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d^3*g

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 14.8107, size = 1966, normalized size = 5.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[-1/96*(15*(6*(4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (32*c^3*d^4 - 36*b*c^2*d^3*e + 12*b^2*c*d^2*e^
2 - b^3*d*e^3)*g + (6*(4*c^3*d^2*e^2 - 4*b*c^2*d*e^3 + b^2*c*e^4)*f - (32*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 12*b^
2*c*d*e^3 - b^3*e^4)*g)*x)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 + 4*sqrt
(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*(8*c^3*e^3*g*x^3 + 2*(6*c^3*e^3*f - (20*c
^3*d*e^2 - 13*b*c^2*e^3)*g)*x^2 - 6*(48*c^3*d^2*e - 41*b*c^2*d*e^2 + 8*b^2*c*e^3)*f + (376*c^3*d^3 - 352*b*c^2
*d^2*e + 81*b^2*c*d*e^2)*g - (6*(14*c^3*d*e^2 - 9*b*c^2*e^3)*f - (136*c^3*d^2*e - 134*b*c^2*d*e^2 + 33*b^2*c*e
^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c*e^3*x + c*d*e^2), 1/48*(15*(6*(4*c^3*d^3*e - 4*b*c^2*
d^2*e^2 + b^2*c*d*e^3)*f - (32*c^3*d^4 - 36*b*c^2*d^3*e + 12*b^2*c*d^2*e^2 - b^3*d*e^3)*g + (6*(4*c^3*d^2*e^2
- 4*b*c^2*d*e^3 + b^2*c*e^4)*f - (32*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 12*b^2*c*d*e^3 - b^3*e^4)*g)*x)*sqrt(c)*ar
ctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2
+ b*c*d*e)) + 2*(8*c^3*e^3*g*x^3 + 2*(6*c^3*e^3*f - (20*c^3*d*e^2 - 13*b*c^2*e^3)*g)*x^2 - 6*(48*c^3*d^2*e -
41*b*c^2*d*e^2 + 8*b^2*c*e^3)*f + (376*c^3*d^3 - 352*b*c^2*d^2*e + 81*b^2*c*d*e^2)*g - (6*(14*c^3*d*e^2 - 9*b*
c^2*e^3)*f - (136*c^3*d^2*e - 134*b*c^2*d*e^2 + 33*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)
)/(c*e^3*x + c*d*e^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**4,x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

Timed out